🎬the Ideal Helmholtz Case and the Coupled Boundary Case

The two demos illustrate how the mathematical boundary conditions of a volume dictate the physical independence of different flow types. In Demo 1, the solenoidal field is strictly orthogonal to the surface normal, satisfying the conditions for the Helmholtz decomposition; consequently, the cross-term integral vanishes, and the total kinetic energy is perfectly additive. In contrast, Demo 2 introduces "boundary leakage," where the solenoidal field is forced to align partially with the irrotational gradient. This violation breaks the orthogonality, creating an interaction energy that causes the total energy to deviate from the sum of its parts. Together, these simulations prove that energy conservation between potential and vortex flows is not an inherent property of the fields themselves, but a direct consequence of the boundary constraints.

🎬Narrated Video

🪜The Energetic Orthogonality of Solenoidal and Irrotational Fields

Description

Orthogonal System (The Problem Solution & Demo 1): This state represents the ideal mathematical scenario where the vector field w is strictly tangent to the boundary ( $w \cdot n = 0$ ). In this state, the integral I vanishes, meaning the irrotational and solenoidal components are energy orthogonal. The total kinetic energy of the system is simply the sum of its individual parts, with no "cross-talk" or interference.
Non-Orthogonal System (Demo 2 & 3): This state occurs when "boundary leakage" is introduced, forcing w to have a component parallel to the surface normal ( $w \cdot n \neq 0$ ). The integral I becomes a non-zero value, acting as interaction energy. In this state, the two types of motion "communicate" through the boundary, and the total energy no longer equals the sum of the parts.
Transitions: The system moves between these states based on the boundary constraints. Enforcing the condition $w \cdot n = 0$ maintains the orthogonal state, while violating it—such as in open pipes or antennas where energy "leaks" out—shifts the system into a non-orthogonal, coupled state.